Graphs, DSA Complete Syllabus

Course Curriculum: Mastering Graph Theory and Algorithms

This comprehensive course is designed to take learners from foundational concepts to advanced topics in graph theory and algorithms. It covers every aspect of graph data structures, algorithms, and their applications, ensuring mastery for academic, professional, and competitive programming purposes.

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Module 1: Introduction to Graph Theory

1. What is a Graph?
   • Definition and basic terminology.
   • Components: vertices, edges, degree, adjacency.
   • Types of graphs: directed, undirected, weighted, unweighted, cyclic, acyclic.
2. Mathematical Representation of Graphs
   • Adjacency matrix.
   • Adjacency list.
   • Incidence matrix.
3. Real-world Applications of Graphs
   • Networking.
   • Social media connections.
   • Transportation systems.
4. Comparison with Other Data Structures
   • Graphs vs Trees.
   • Graphs vs Sets and Lists.

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Module 2: Graph Representation and Implementation

1. Graph Storage Techniques
   • Sparse vs dense graphs.
   • Memory-efficient representations.
2. Graph Input and Visualization
   • Input formats for graphs.
   • Visualizing graphs using tools like Graphviz.
3. Weighted Graphs
   • Representing weights in adjacency lists and matrices.
4. Dynamic Graphs
   • Handling addition and removal of edges and vertices.
   • Use cases of dynamic graphs.

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Module 3: Graph Traversal Techniques

1. Depth-First Search (DFS)
   • Recursive and iterative implementations.
   • Applications: detecting cycles, pathfinding.
2. Breadth-First Search (BFS)
   • Iterative implementation using queues.
   • Applications: shortest paths in unweighted graphs, level-order traversal.
3. Bidirectional Search
   • Concept and implementation.
   • Efficiency comparison with BFS.
4. Traversal Use Cases
   • Solving maze problems.
   • Finding connected components.

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Module 4: Fundamental Graph Properties

1. Connectivity
   • Strongly connected components (SCCs).
   • Weakly connected components.
   • Algorithms for finding SCCs: Tarjan's and Kosaraju’s algorithms.
2. Eulerian and Hamiltonian Paths
   • Definitions and properties.
   • Algorithms for detecting Eulerian paths/cycles.
   • Hamiltonian cycle challenges and solutions.
3. Graph Coloring
   • Vertex coloring.
   • Chromatic number and applications.
   • Greedy coloring algorithms.

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Module 5: Shortest Path Algorithms

1. Single-source Shortest Path
   • Dijkstra’s Algorithm.
   • Bellman-Ford Algorithm.
2. All-pairs Shortest Path
   • Floyd-Warshall Algorithm.
   • Johnson’s Algorithm.
3. A* Search Algorithm
   • Heuristic-based shortest path finding.
   • Applications in gaming and navigation.
4. Applications of Shortest Path Algorithms
   • Routing in networks.
   • Optimizing transportation systems.

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Module 6: Minimum Spanning Tree (MST) Algorithms

1. What is an MST?
   • Definition and properties.
   • Real-world applications.
2. Kruskal’s Algorithm
   • Greedy approach and implementation.
   • Union-Find data structure.
3. Prim’s Algorithm
   • Priority queue-based implementation.
   • Comparison with Kruskal’s algorithm.
4. Boruvka’s Algorithm
   • Efficient MST construction for dense graphs.
5. Applications of MST
   • Network design.
   • Cost optimization problems.

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Module 7: Advanced Graph Algorithms

1. Topological Sorting
   • Kahn’s Algorithm.
   • DFS-based approach.
   • Applications in dependency resolution.
2. Cycle Detection
   • Directed and undirected graphs.
   • Algorithms for detecting cycles.
3. Maximum Flow Algorithms
   • Ford-Fulkerson method.
   • Edmonds-Karp implementation.
   • Dinic’s Algorithm for efficiency.
4. Bipartite Graphs
   • Testing for bipartiteness.
   • Matching and applications.
5. Strongly Connected Components
   • Kosaraju’s and Tarjan’s algorithms.
   • Use cases in software dependency analysis.

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Module 8: Advanced Graph Theory

1. Graph Isomorphism
   • Concept and challenges.
   • Algorithms for isomorphism testing.
2. Planar Graphs
   • Properties and representations.
   • Kuratowski's theorem.
3. Tree-based Graphs
   • Spanning trees and arborescences.
   • Center and centroid of trees.
4. Graph Metrics
   • Centrality measures: degree, closeness, betweenness.
   • Diameter, radius, and eccentricity.

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Module 9: Dynamic and Specialized Graph Algorithms

1. Dynamic Graph Algorithms
   • Dynamic shortest paths.
   • Incremental and decremental graph updates.
2. Graph Matching
   • Maximum bipartite matching (Hungarian algorithm).
   • Applications in job assignment.
3. Graph Partitioning
   • Partitioning algorithms for load balancing.
   • Applications in distributed systems.
4. Randomized Graph Algorithms
   • Monte Carlo and Las Vegas methods.

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Module 10: Graphs in Competitive Programming

1. Common Graph Problems
   • Problem-solving on platforms like Codeforces, LeetCode, and HackerRank.
2. Optimizing Graph Solutions
   • Space and time trade-offs.
   • Efficient data structures for graph problems.
3. Common Mistakes and Debugging
   • Detecting and fixing traversal and connectivity issues.

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Module 11: Graph Applications in Real-world Scenarios

1. Networking
   • Shortest path routing protocols.
   • Bandwidth optimization.
2. Social Networks
   • Community detection.
   • Graph-based influence spread analysis.
3. Transportation and Logistics
   • Route optimization.
   • Traffic flow analysis.
4. Artificial Intelligence
   • Search algorithms in game AI.
   • Graph-based knowledge representation.
5. Biological Networks
   • Protein interaction networks.
   • Gene regulatory networks.

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Module 12: Practical Projects and Case Studies

1. Dynamic Road Network
   • Building a graph-based traffic management system.
2. Social Media Graph
   • Analyzing connections and influence in social networks.
3. Recommendation System
   • Graph-based product recommendation using collaborative filtering.
4. Distributed System Design
   • Graph partitioning for scalable systems.

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Module 13: Advanced Topics and Research Areas

1. Spectral Graph Theory
   • Eigenvalues and eigenvectors in graphs.
   • Laplacian matrix applications.
2. Graph Neural Networks
   • Basics of GNNs and applications in machine learning.
3. Quantum Graph Algorithms
   • Emerging algorithms in quantum computing.
4. Graph Compression
   • Techniques for efficient storage and processing of large graphs.

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Module 14: Final Assessment and Mastery

1. Comprehensive Coding Projects
   • Design and implement a graph library.
   • Build a real-world application leveraging advanced graph algorithms.
2. Capstone Project
   • Solve a large-scale, real-world problem using graph theory.
3. Final Examination
   • Theory and practical coding assessment.
4. Feedback and Next Steps
   • Personalized feedback for improvement.
   • Recommendations for advanced study (e.g., graph databases, hypergraphs).

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This curriculum provides a thorough understanding of graphs and their algorithms, making it ideal for learners aiming to excel in academic research, software engineering, or competitive programming.